!! Module to put all of newuoa-Fortran77 code of Professor Michael J.D. Powell
!! in one file and 
!! provide a Subroutine with a 
!! Fortran 95 interface for the top level routine.
!! Newuoa is an optimization routine for unconstrained optimization
!! without using derivatives of the objective function.
!! Newuoa consideres an objective function evaluation to be expensive.

!! This module is distributed under the same terms as Professor Michael Powells
!! newuoa. Basically GNU and you can use it freely.


MODULE newuoa_module


   private
   public :: minimize_with_newuoa

  contains

   !! The routine minimizes CALFUN in X such that the function is
   !! minimized without using derivatives. 
   !! - SUBROUTINE CALFUN (i_X,o_F) is the function to be minimized without using derivatives.
   !!     It must set o_F (output) to
   !!     the value of the objective function for the (input) 
   !!     variables i_X(1),i_X(2),...,i_X(N).
   !! - X: on entry initial x where search starts. On exit best found
   !!      vector x that minimizes CALFUN.
   !! - RHOBEG and RHOEND must be set to the initial and final values of a trust
   !!      region radius, so both must be positive with RHOEND<=RHOBEG. Typically
   !!      RHOBEG should be about one tenth of the greatest expected change to a
   !!      variable, and RHOEND should indicate the accuracy that is required in
   !!      the final values of the variables.
   !! - The value of IPRINT should be set to 0, 1, 2 or 3, which controls the
   !!      amount of printing. Specifically, there is no output if IPRINT=0 and
   !!      there is output only at the return if IPRINT=1. Otherwise, each new
   !!      value of RHO is printed, with the best vector of variables so far and
   !!      the corresponding value of the objective function. Further, each new
   !!      value of F with its variables are output if IPRINT=3. 
   !!      Default IPRINT=0 if i_opt_IPRINT is not set.
   !! - NPT is the number of interpolation conditions. Its value must be in the
   !!      interval [N+2,(N+1)(N+2)/2] with N=size(X).
   !!      if i_opt_NPT is not set Default NPT=2*N+1.
   !! - MAXFUN: maximal number of calls to CALFUN.
   !!      default value MAXFUN=10000 if i_opt_MAXFUN is not set.

   SUBROUTINE minimize_with_newuoa(CALFUN, X, i_RHOBEG, i_RHOEND, &
                              i_opt_NPT, i_opt_IPRINT, i_opt_MAXFUN)
      implicit none
      interface
         subroutine CALFUN(i_x, o_f)
            real(kind=8), intent(in), dimension(:) :: i_x
            real(kind=8), intent(out) :: o_f
         end subroutine
      end interface
      real(kind=8), intent(inout), dimension(:) :: X
      real(kind=8), intent(in) :: i_RHOBEG, i_RHOEND
      integer, intent(in), optional :: i_opt_NPT, i_opt_IPRINT, i_opt_MAXFUN
      
      integer :: NPT, IPRINT, MAXFUN, N
      real(kind=8), dimension(:), allocatable :: W
      
      N = size(X)
      
      if(present(i_opt_NPT)) then
         NPT = i_opt_NPT
      else
         NPT = 2*N + 1
      end if
      
      if(present(i_opt_IPRINT)) then
         IPRINT = i_opt_IPRINT
      else
         IPRINT = 0
      end if
      
      if(present(i_opt_MAXFUN)) then
         MAXFUN = i_opt_MAXFUN
      else
         MAXFUN = 10000
      end if
      
      allocate( W( (NPT+13)*(NPT+N)+3*N*(N+3)/2 ) )
      
      call NEWUOA( CALFUN,N,NPT,X,i_RHOBEG,i_RHOEND, &
                              IPRINT, MAXFUN, W )
      deallocate( W )
      
   END SUBROUTINE minimize_with_newuoa
      

  
      SUBROUTINE NEWUOA(CALFUN,N,NPT,X,RHOBEG,RHOEND,IPRINT,MAXFUN,W)
         implicit none
         integer, intent(in) :: N, NPT, IPRINT, MAXFUN
         real(kind=8), intent(inout) :: X(:), W(*)
         real(kind=8), intent(in) :: RHOBEG,RHOEND
         interface
            subroutine CALFUN(i_x,o_f)
               real(kind=8), intent(in), dimension(:) :: i_x
               real(kind=8), intent(out) :: o_f
            end subroutine
         end interface
      
      
      !local variables:
         integer :: NP, NPTM, NDIM, IXB, IXO, IXN, IXP, IFV, IGQ, &
                    IHQ, IPQ, IBMAT, IZMAT, ID, IVL, IW 
      
!
!     This subroutine seeks the least value of a function of many variables,
!     by a trust region method that forms quadratic models by interpolation.
!     There can be some freedom in the interpolation conditions, which is
!     taken up by minimizing the Frobenius norm of the change to the second
!     derivative of the quadratic model, beginning with a zero matrix. The
!     arguments of the subroutine are as follows.

!     N must be set to the number of variables and must be at least two.
!     NPT is the number of interpolation conditions. Its value must be in the
!       interval [N+2,(N+1)(N+2)/2].
!     Initial values of the variables must be set in X(1),X(2),...,X(N). They
!       will be changed to the values that give the least calculated F.
!     RHOBEG and RHOEND must be set to the initial and final values of a trust
!       region radius, so both must be positive with RHOEND<=RHOBEG. Typically
!       RHOBEG should be about one tenth of the greatest expected change to a
!       variable, and RHOEND should indicate the accuracy that is required in
!       the final values of the variables.
!     The value of IPRINT should be set to 0, 1, 2 or 3, which controls the
!       amount of printing. Specifically, there is no output if IPRINT=0 and
!       there is output only at the return if IPRINT=1. Otherwise, each new
!       value of RHO is printed, with the best vector of variables so far and
!       the corresponding value of the objective function. Further, each new
!       value of F with its variables are output if IPRINT=3.
!     MAXFUN must be set to an upper bound on the number of calls of CALFUN.
!     The array W will be used for working space. Its length must be at least
!     (NPT+13)*(NPT+N)+3*N*(N+3)/2.

!     SUBROUTINE CALFUN (X,F) must be provided by the user. It must set F to
!     the value of the objective function for the variables X(1),X(2),...,X(N).

!     Partition the working space array, so that different parts of it can be
!     treated separately by the subroutine that performs the main calculation.

      NP=N+1
      NPTM=NPT-NP
      IF (NPT .LT. N+2 .OR. NPT .GT. ((N+2)*NP)/2) THEN
          PRINT 10
   10     FORMAT (/4X,'Return from NEWUOA because NPT is not in', &
           ' the required interval')
          GO TO 20
      END IF
      NDIM=NPT+N
      IXB=1
      IXO=IXB+N
      IXN=IXO+N
      IXP=IXN+N
      IFV=IXP+N*NPT
      IGQ=IFV+NPT
      IHQ=IGQ+N
      IPQ=IHQ+(N*NP)/2
      IBMAT=IPQ+NPT
      IZMAT=IBMAT+NDIM*N
      ID=IZMAT+NPT*NPTM
      IVL=ID+N
      IW=IVL+NDIM

!     The above settings provide a partition of W for subroutine NEWUOB.
!     The partition requires the first NPT*(NPT+N)+5*N*(N+3)/2 elements of
!     W plus the space that is needed by the last array of NEWUOB.

      CALL NEWUOB(CALFUN, N,NPT,X,RHOBEG,RHOEND,IPRINT,MAXFUN,W(IXB),  &
        W(IXO),W(IXN),W(IXP),W(IFV),W(IGQ),W(IHQ),W(IPQ),W(IBMAT), &
        W(IZMAT),NDIM,W(ID),W(IVL),W(IW))
   20 RETURN
      END SUBROUTINE NEWUOA

  
  
      SUBROUTINE NEWUOB(CALFUN,N,NPT,X,RHOBEG,RHOEND,IPRINT,MAXFUN,XBASE, &
        XOPT,XNEW,XPT,FVAL,GQ,HQ,PQ,BMAT,ZMAT,NDIM,D,VLAG,W)

        implicit none
        integer, intent(in) :: N, NPT, IPRINT,MAXFUN, NDIM
        real(kind=8), intent(inout), dimension(:) :: X
        real(kind=8), intent(in) :: RHOBEG, RHOEND
        real(kind=8), intent(inout) :: XBASE(*), XOPT(*),XNEW(*),  &
                        FVAL(*), GQ(*), HQ(*), PQ(*), D(*), VLAG(*), W(*)
        real(kind=8), intent(inout) :: BMAT(NDIM,*),ZMAT(NPT,*), XPT(NPT,*)
        
         interface
            subroutine CALFUN(i_x,o_f)
               real(kind=8), intent(in), dimension(:) :: i_x
               real(kind=8), intent(out) :: o_f
            end subroutine
         end interface
         
      !local variables
      integer :: NP, NH,NPTM,NFTEST, NF, NFM, NFMM, ITEMP, JPT, IPT, &
                  IH, IDZ, ITEST, NFSAV, KNEW, I, IP, J, JP, K, KSAVE, &
                  KTEMP, KOPT
      real(kind=8) :: HALF, ONE, TENTH, ZERO, RHOSQ, RECIP,RECIQ, XIPT, &
                     XJPT, FBEG, FOPT, RHO, DELTA, DIFFA, DIFFB, &
                     XOPTSQ, DSQ, DNORM, RATIO, TEMP, CRVMIN, TEMPQ, &
                     ALPHA, BETA, BSUM, DETRAT, DIFF, DIFFC, DISTSQ, &
                     DX, F, FSAVE, GISQ, GQSQ, HDIAG, SUM, SUMA, SUMB, &
                     DSTEP, SUMZ, VQUAD
       
!     The arguments N, NPT, X, RHOBEG, RHOEND, IPRINT and MAXFUN are identical
!       to the corresponding arguments in SUBROUTINE NEWUOA.
!     XBASE will hold a shift of origin that should reduce the contributions
!       from rounding errors to values of the model and Lagrange functions.
!     XOPT will be set to the displacement from XBASE of the vector of
!       variables that provides the least calculated F so far.
!     XNEW will be set to the displacement from XBASE of the vector of
!       variables for the current calculation of F.
!     XPT will contain the interpolation point coordinates relative to XBASE.
!     FVAL will hold the values of F at the interpolation points.
!     GQ will hold the gradient of the quadratic model at XBASE.
!     HQ will hold the explicit second derivatives of the quadratic model.
!     PQ will contain the parameters of the implicit second derivatives of
!       the quadratic model.
!     BMAT will hold the last N columns of H.
!     ZMAT will hold the factorization of the leading NPT by NPT submatrix of
!       H, this factorization being ZMAT times Diag(DZ) times ZMAT^T, where
!       the elements of DZ are plus or minus one, as specified by IDZ.
!     NDIM is the first dimension of BMAT and has the value NPT+N.
!     D is reserved for trial steps from XOPT.
!     VLAG will contain the values of the Lagrange functions at a new point X.
!       They are part of a product that requires VLAG to be of length NDIM.
!     The array W will be used for working space. Its length must be at least
!       10*NDIM = 10*(NPT+N).

!     Set some constants.

      HALF=0.5D0
      ONE=1.0D0
      TENTH=0.1D0
      ZERO=0.0D0
      NP=N+1
      NH=(N*NP)/2
      NPTM=NPT-NP
      NFTEST=MAX0(MAXFUN,1)

!     Set the initial elements of XPT, BMAT, HQ, PQ and ZMAT to zero.

      DO 20 J=1,N
      XBASE(J)=X(J)
      DO 10 K=1,NPT
   10 XPT(K,J)=ZERO
      DO 20 I=1,NDIM
   20 BMAT(I,J)=ZERO
      DO 30 IH=1,NH
   30 HQ(IH)=ZERO
      DO 40 K=1,NPT
      PQ(K)=ZERO
      DO 40 J=1,NPTM
   40 ZMAT(K,J)=ZERO

!     Begin the initialization procedure. NF becomes one more than the number
!     of function values so far. The coordinates of the displacement of the
!     next initial interpolation point from XBASE are set in XPT(NF,.).

      RHOSQ=RHOBEG*RHOBEG
      RECIP=ONE/RHOSQ
      RECIQ=DSQRT(HALF)/RHOSQ
      NF=0
   50 NFM=NF
      NFMM=NF-N
      NF=NF+1
      IF (NFM .LE. 2*N) THEN
          IF (NFM .GE. 1 .AND. NFM .LE. N) THEN
              XPT(NF,NFM)=RHOBEG
          ELSE IF (NFM .GT. N) THEN
              XPT(NF,NFMM)=-RHOBEG
          END IF
      ELSE
          ITEMP=(NFMM-1)/N
          JPT=NFM-ITEMP*N-N
          IPT=JPT+ITEMP
          IF (IPT .GT. N) THEN
              ITEMP=JPT
              JPT=IPT-N
              IPT=ITEMP
          END IF
          XIPT=RHOBEG
          IF (FVAL(IPT+NP) .LT. FVAL(IPT+1)) XIPT=-XIPT
          XJPT=RHOBEG
          IF (FVAL(JPT+NP) .LT. FVAL(JPT+1)) XJPT=-XJPT
          XPT(NF,IPT)=XIPT
          XPT(NF,JPT)=XJPT
      END IF

!     Calculate the next value of F, label 70 being reached immediately
!     after this calculation. The least function value so far and its index
!     are required.

      DO 60 J=1,N
   60 X(J)=XPT(NF,J)+XBASE(J)
      GOTO 310
   70 FVAL(NF)=F
      IF (NF .EQ. 1) THEN
          FBEG=F
          FOPT=F
          KOPT=1
      ELSE IF (F .LT. FOPT) THEN
          FOPT=F
          KOPT=NF
      END IF

!     Set the nonzero initial elements of BMAT and the quadratic model in
!     the cases when NF is at most 2*N+1.

      IF (NFM .LE. 2*N) THEN
          IF (NFM .GE. 1 .AND. NFM .LE. N) THEN
              GQ(NFM)=(F-FBEG)/RHOBEG
              IF (NPT .LT. NF+N) THEN
                  BMAT(1,NFM)=-ONE/RHOBEG
                  BMAT(NF,NFM)=ONE/RHOBEG
                  BMAT(NPT+NFM,NFM)=-HALF*RHOSQ
              END IF
          ELSE IF (NFM .GT. N) THEN
              BMAT(NF-N,NFMM)=HALF/RHOBEG
              BMAT(NF,NFMM)=-HALF/RHOBEG
              ZMAT(1,NFMM)=-RECIQ-RECIQ
              ZMAT(NF-N,NFMM)=RECIQ
              ZMAT(NF,NFMM)=RECIQ
              IH=(NFMM*(NFMM+1))/2
              TEMP=(FBEG-F)/RHOBEG
              HQ(IH)=(GQ(NFMM)-TEMP)/RHOBEG
              GQ(NFMM)=HALF*(GQ(NFMM)+TEMP)
          END IF

!     Set the off-diagonal second derivatives of the Lagrange functions and
!     the initial quadratic model.

      ELSE
          IH=(IPT*(IPT-1))/2+JPT
          IF (XIPT .LT. ZERO) IPT=IPT+N
          IF (XJPT .LT. ZERO) JPT=JPT+N
          ZMAT(1,NFMM)=RECIP
          ZMAT(NF,NFMM)=RECIP
          ZMAT(IPT+1,NFMM)=-RECIP
          ZMAT(JPT+1,NFMM)=-RECIP
          HQ(IH)=(FBEG-FVAL(IPT+1)-FVAL(JPT+1)+F)/(XIPT*XJPT)
      END IF
      IF (NF .LT. NPT) GOTO 50

!     Begin the iterative procedure, because the initial model is complete.

      RHO=RHOBEG
      DELTA=RHO
      IDZ=1
      DIFFA=ZERO
      DIFFB=ZERO
      ITEST=0
      XOPTSQ=ZERO
      DO 80 I=1,N
      XOPT(I)=XPT(KOPT,I)
   80 XOPTSQ=XOPTSQ+XOPT(I)**2
   90 NFSAV=NF

!     Generate the next trust region step and test its length. Set KNEW
!     to -1 if the purpose of the next F will be to improve the model.

  100 KNEW=0
      CALL TRSAPP (N,NPT,XOPT,XPT,GQ,HQ,PQ,DELTA,D,W,W(NP), &
        W(NP+N),W(NP+2*N),CRVMIN)
      DSQ=ZERO
      DO 110 I=1,N
  110 DSQ=DSQ+D(I)**2
      DNORM=DMIN1(DELTA,DSQRT(DSQ))
      IF (DNORM .LT. HALF*RHO) THEN
          KNEW=-1
          DELTA=TENTH*DELTA
          RATIO=-1.0D0
          IF (DELTA .LE. 1.5D0*RHO) DELTA=RHO
          IF (NF .LE. NFSAV+2) GOTO 460
          TEMP=0.125D0*CRVMIN*RHO*RHO
          IF (TEMP .LE. DMAX1(DIFFA,DIFFB,DIFFC)) GOTO 460
          GOTO 490
      END IF

!     Shift XBASE if XOPT may be too far from XBASE. First make the changes
!     to BMAT that do not depend on ZMAT.

  120 IF (DSQ .LE. 1.0D-3*XOPTSQ) THEN
          TEMPQ=0.25D0*XOPTSQ
          DO 140 K=1,NPT
          SUM=ZERO
          DO 130 I=1,N
  130     SUM=SUM+XPT(K,I)*XOPT(I)
          TEMP=PQ(K)*SUM
          SUM=SUM-HALF*XOPTSQ
          W(NPT+K)=SUM
          DO 140 I=1,N
          GQ(I)=GQ(I)+TEMP*XPT(K,I)
          XPT(K,I)=XPT(K,I)-HALF*XOPT(I)
          VLAG(I)=BMAT(K,I)
          W(I)=SUM*XPT(K,I)+TEMPQ*XOPT(I)
          IP=NPT+I
          DO 140 J=1,I
  140     BMAT(IP,J)=BMAT(IP,J)+VLAG(I)*W(J)+W(I)*VLAG(J)

!     Then the revisions of BMAT that depend on ZMAT are calculated.

          DO 180 K=1,NPTM
          SUMZ=ZERO
          DO 150 I=1,NPT
          SUMZ=SUMZ+ZMAT(I,K)
  150     W(I)=W(NPT+I)*ZMAT(I,K)
          DO 170 J=1,N
          SUM=TEMPQ*SUMZ*XOPT(J)
          DO 160 I=1,NPT
  160     SUM=SUM+W(I)*XPT(I,J)
          VLAG(J)=SUM
          IF (K .LT. IDZ) SUM=-SUM
          DO 170 I=1,NPT
  170     BMAT(I,J)=BMAT(I,J)+SUM*ZMAT(I,K)
          DO 180 I=1,N
          IP=I+NPT
          TEMP=VLAG(I)
          IF (K .LT. IDZ) TEMP=-TEMP
          DO 180 J=1,I
  180     BMAT(IP,J)=BMAT(IP,J)+TEMP*VLAG(J)

!     The following instructions complete the shift of XBASE, including
!     the changes to the parameters of the quadratic model.

          IH=0
          DO 200 J=1,N
          W(J)=ZERO
          DO 190 K=1,NPT
          W(J)=W(J)+PQ(K)*XPT(K,J)
  190     XPT(K,J)=XPT(K,J)-HALF*XOPT(J)
          DO 200 I=1,J
          IH=IH+1
          IF (I .LT. J) GQ(J)=GQ(J)+HQ(IH)*XOPT(I)
          GQ(I)=GQ(I)+HQ(IH)*XOPT(J)
          HQ(IH)=HQ(IH)+W(I)*XOPT(J)+XOPT(I)*W(J)
  200     BMAT(NPT+I,J)=BMAT(NPT+J,I)
          DO 210 J=1,N
          XBASE(J)=XBASE(J)+XOPT(J)
  210     XOPT(J)=ZERO
          XOPTSQ=ZERO
      END IF

!     Pick the model step if KNEW is positive. A different choice of D
!     may be made later, if the choice of D by BIGLAG causes substantial
!     cancellation in DENOM.

      IF (KNEW .GT. 0) THEN
          CALL BIGLAG (N,NPT,XOPT,XPT,BMAT,ZMAT,IDZ,NDIM,KNEW,DSTEP, &
            D,ALPHA,VLAG,VLAG(NPT+1),W,W(NP),W(NP+N))
      END IF

!     Calculate VLAG and BETA for the current choice of D. The first NPT
!     components of W_check will be held in W.

      DO 230 K=1,NPT
      SUMA=ZERO
      SUMB=ZERO
      SUM=ZERO
      DO 220 J=1,N
      SUMA=SUMA+XPT(K,J)*D(J)
      SUMB=SUMB+XPT(K,J)*XOPT(J)
  220 SUM=SUM+BMAT(K,J)*D(J)
      W(K)=SUMA*(HALF*SUMA+SUMB)
  230 VLAG(K)=SUM
      BETA=ZERO
      DO 250 K=1,NPTM
      SUM=ZERO
      DO 240 I=1,NPT
  240 SUM=SUM+ZMAT(I,K)*W(I)
      IF (K .LT. IDZ) THEN
          BETA=BETA+SUM*SUM
          SUM=-SUM
      ELSE
          BETA=BETA-SUM*SUM
      END IF
      DO 250 I=1,NPT
  250 VLAG(I)=VLAG(I)+SUM*ZMAT(I,K)
      BSUM=ZERO
      DX=ZERO
      DO 280 J=1,N
      SUM=ZERO
      DO 260 I=1,NPT
  260 SUM=SUM+W(I)*BMAT(I,J)
      BSUM=BSUM+SUM*D(J)
      JP=NPT+J
      DO 270 K=1,N
  270 SUM=SUM+BMAT(JP,K)*D(K)
      VLAG(JP)=SUM
      BSUM=BSUM+SUM*D(J)
  280 DX=DX+D(J)*XOPT(J)
      BETA=DX*DX+DSQ*(XOPTSQ+DX+DX+HALF*DSQ)+BETA-BSUM
      VLAG(KOPT)=VLAG(KOPT)+ONE

!     If KNEW is positive and if the cancellation in DENOM is unacceptable,
!     then BIGDEN calculates an alternative model step, XNEW being used for
!     working space.

      IF (KNEW .GT. 0) THEN
          TEMP=ONE+ALPHA*BETA/VLAG(KNEW)**2
          IF (DABS(TEMP) .LE. 0.8D0) THEN
              CALL BIGDEN (N,NPT,XOPT,XPT,BMAT,ZMAT,IDZ,NDIM,KOPT, &
                KNEW,D,W,VLAG,BETA,XNEW,W(NDIM+1),W(6*NDIM+1))
          END IF
      END IF

!     Calculate the next value of the objective function.

  290 DO 300 I=1,N
      XNEW(I)=XOPT(I)+D(I)
  300 X(I)=XBASE(I)+XNEW(I)
      NF=NF+1
  310 IF (NF .GT. NFTEST) THEN
          NF=NF-1
          IF (IPRINT .GT. 0) PRINT 320
  320     FORMAT (/4X,'Return from NEWUOA because CALFUN has been', &
            ' called MAXFUN times.')
          GOTO 530
      END IF
      CALL CALFUN(X,F)
      IF (IPRINT .EQ. 3) THEN
          PRINT 330, NF,F,(X(I),I=1,N)
  330      FORMAT (/4X,'Function number',I6,'    F =',1PD18.10, &
             '    The corresponding X is:'/(2X,5D15.6))
      END IF
      IF (NF .LE. NPT) GOTO 70
      IF (KNEW .EQ. -1) GOTO 530

!     Use the quadratic model to predict the change in F due to the step D,
!     and set DIFF to the error of this prediction.

      VQUAD=ZERO
      IH=0
      DO 340 J=1,N
      VQUAD=VQUAD+D(J)*GQ(J)
      DO 340 I=1,J
      IH=IH+1
      TEMP=D(I)*XNEW(J)+D(J)*XOPT(I)
      IF (I .EQ. J) TEMP=HALF*TEMP
  340 VQUAD=VQUAD+TEMP*HQ(IH)
      DO 350 K=1,NPT
  350 VQUAD=VQUAD+PQ(K)*W(K)
      DIFF=F-FOPT-VQUAD
      DIFFC=DIFFB
      DIFFB=DIFFA
      DIFFA=DABS(DIFF)
      IF (DNORM .GT. RHO) NFSAV=NF

!     Update FOPT and XOPT if the new F is the least value of the objective
!     function so far. The branch when KNEW is positive occurs if D is not
!     a trust region step.

      FSAVE=FOPT
      IF (F .LT. FOPT) THEN
          FOPT=F
          XOPTSQ=ZERO
          DO 360 I=1,N
          XOPT(I)=XNEW(I)
  360     XOPTSQ=XOPTSQ+XOPT(I)**2
      END IF
      KSAVE=KNEW
      IF (KNEW .GT. 0) GOTO 410

!     Pick the next value of DELTA after a trust region step.

      IF (VQUAD .GE. ZERO) THEN
          IF (IPRINT .GT. 0) PRINT 370
  370     FORMAT (/4X,'Return from NEWUOA because a trust', &
            ' region step has failed to reduce Q.')
          GOTO 530
      END IF
      RATIO=(F-FSAVE)/VQUAD
      IF (RATIO .LE. TENTH) THEN
          DELTA=HALF*DNORM
      ELSE IF (RATIO .LE. 0.7D0) THEN
          DELTA=DMAX1(HALF*DELTA,DNORM)
      ELSE
          DELTA=DMAX1(HALF*DELTA,DNORM+DNORM)
      END IF
      IF (DELTA .LE. 1.5D0*RHO) DELTA=RHO

!     Set KNEW to the index of the next interpolation point to be deleted.

      RHOSQ=DMAX1(TENTH*DELTA,RHO)**2
      KTEMP=0
      DETRAT=ZERO
      IF (F .GE. FSAVE) THEN
          KTEMP=KOPT
          DETRAT=ONE
      END IF
      DO 400 K=1,NPT
      HDIAG=ZERO
      DO 380 J=1,NPTM
      TEMP=ONE
      IF (J .LT. IDZ) TEMP=-ONE
  380 HDIAG=HDIAG+TEMP*ZMAT(K,J)**2
      TEMP=DABS(BETA*HDIAG+VLAG(K)**2)
      DISTSQ=ZERO
      DO 390 J=1,N
  390 DISTSQ=DISTSQ+(XPT(K,J)-XOPT(J))**2
      IF (DISTSQ .GT. RHOSQ) TEMP=TEMP*(DISTSQ/RHOSQ)**3
      IF (TEMP .GT. DETRAT .AND. K .NE. KTEMP) THEN
          DETRAT=TEMP
          KNEW=K
      END IF
  400 CONTINUE
      IF (KNEW .EQ. 0) GOTO 460

!     Update BMAT, ZMAT and IDZ, so that the KNEW-th interpolation point
!     can be moved. Begin the updating of the quadratic model, starting
!     with the explicit second derivative term.

  410 CALL UPDATE (N,NPT,BMAT,ZMAT,IDZ,NDIM,VLAG,BETA,KNEW,W)
      FVAL(KNEW)=F
      IH=0
      DO 420 I=1,N
      TEMP=PQ(KNEW)*XPT(KNEW,I)
      DO 420 J=1,I
      IH=IH+1
  420 HQ(IH)=HQ(IH)+TEMP*XPT(KNEW,J)
      PQ(KNEW)=ZERO

!     Update the other second derivative parameters, and then the gradient
!     vector of the model. Also include the new interpolation point.

      DO 440 J=1,NPTM
      TEMP=DIFF*ZMAT(KNEW,J)
      IF (J .LT. IDZ) TEMP=-TEMP
      DO 440 K=1,NPT
  440 PQ(K)=PQ(K)+TEMP*ZMAT(K,J)
      GQSQ=ZERO
      DO 450 I=1,N
      GQ(I)=GQ(I)+DIFF*BMAT(KNEW,I)
      GQSQ=GQSQ+GQ(I)**2
  450 XPT(KNEW,I)=XNEW(I)

!     If a trust region step makes a small change to the objective function,
!     then calculate the gradient of the least Frobenius norm interpolant at
!     XBASE, and store it in W, using VLAG for a vector of right hand sides.

      IF (KSAVE .EQ. 0 .AND. DELTA .EQ. RHO) THEN
          IF (DABS(RATIO) .GT. 1.0D-2) THEN
              ITEST=0
          ELSE
              DO 700 K=1,NPT
  700         VLAG(K)=FVAL(K)-FVAL(KOPT)
              GISQ=ZERO
              DO 720 I=1,N
              SUM=ZERO
              DO 710 K=1,NPT
  710         SUM=SUM+BMAT(K,I)*VLAG(K)
              GISQ=GISQ+SUM*SUM
  720         W(I)=SUM

!     Test whether to replace the new quadratic model by the least Frobenius
!     norm interpolant, making the replacement if the test is satisfied.

              ITEST=ITEST+1
              IF (GQSQ .LT. 1.0D2*GISQ) ITEST=0
              IF (ITEST .GE. 3) THEN
                  DO 730 I=1,N
  730             GQ(I)=W(I)
                  DO 740 IH=1,NH
  740             HQ(IH)=ZERO
                  DO 760 J=1,NPTM
                  W(J)=ZERO
                  DO 750 K=1,NPT
  750             W(J)=W(J)+VLAG(K)*ZMAT(K,J)
  760             IF (J .LT. IDZ) W(J)=-W(J)
                  DO 770 K=1,NPT
                  PQ(K)=ZERO
                  DO 770 J=1,NPTM
  770             PQ(K)=PQ(K)+ZMAT(K,J)*W(J)
                  ITEST=0
              END IF
          END IF
      END IF
      IF (F .LT. FSAVE) KOPT=KNEW

!     If a trust region step has provided a sufficient decrease in F, then
!     branch for another trust region calculation. The case KSAVE>0 occurs
!     when the new function value was calculated by a model step.

      IF (F .LE. FSAVE+TENTH*VQUAD) GOTO 100
      IF (KSAVE .GT. 0) GOTO 100

!     Alternatively, find out if the interpolation points are close enough
!     to the best point so far.

      KNEW=0
  460 DISTSQ=4.0D0*DELTA*DELTA
      DO 480 K=1,NPT
      SUM=ZERO
      DO 470 J=1,N
  470 SUM=SUM+(XPT(K,J)-XOPT(J))**2
      IF (SUM .GT. DISTSQ) THEN
          KNEW=K
          DISTSQ=SUM
      END IF
  480 CONTINUE

!     If KNEW is positive, then set DSTEP, and branch back for the next
!     iteration, which will generate a "model step".

      IF (KNEW .GT. 0) THEN
          DSTEP=DMAX1(DMIN1(TENTH*DSQRT(DISTSQ),HALF*DELTA),RHO)
          DSQ=DSTEP*DSTEP
          GOTO 120
      END IF
      IF (RATIO .GT. ZERO) GOTO 100
      IF (DMAX1(DELTA,DNORM) .GT. RHO) GOTO 100

!     The calculations with the current value of RHO are complete. Pick the
!     next values of RHO and DELTA.

  490 IF (RHO .GT. RHOEND) THEN
          DELTA=HALF*RHO
          RATIO=RHO/RHOEND
          IF (RATIO .LE. 16.0D0) THEN
              RHO=RHOEND
          ELSE IF (RATIO .LE. 250.0D0) THEN
              RHO=DSQRT(RATIO)*RHOEND
          ELSE
              RHO=TENTH*RHO
          END IF
          DELTA=DMAX1(DELTA,RHO)
          IF (IPRINT .GE. 2) THEN
              IF (IPRINT .GE. 3) PRINT 500
  500         FORMAT (5X)
              PRINT 510, RHO,NF
  510         FORMAT (/4X,'New RHO =',1PD11.4,5X,'Number of', &
                ' function values =',I6)
              PRINT 520, FOPT,(XBASE(I)+XOPT(I),I=1,N)
  520         FORMAT (4X,'Least value of F =',1PD23.15,9X, &
                'The corresponding X is:'/(2X,5D15.6))
          END IF
          GOTO 90
      END IF

!     Return from the calculation, after another Newton-Raphson step, if
!     it is too short to have been tried before.

      IF (KNEW .EQ. -1) GOTO 290
  530 IF (FOPT .LE. F) THEN
          DO 540 I=1,N
  540     X(I)=XBASE(I)+XOPT(I)
          F=FOPT
      END IF
      IF (IPRINT .GE. 1) THEN
          PRINT 550, NF
  550     FORMAT (/4X,'At the return from NEWUOA',5X, &
            'Number of function values =',I6)
          PRINT 520, F,(X(I),I=1,N)
      END IF
      RETURN
      END SUBROUTINE NEWUOB

  
  
      SUBROUTINE BIGDEN (N,NPT,XOPT,XPT,BMAT,ZMAT,IDZ,NDIM,KOPT, &
        KNEW,D,W,VLAG,BETA,S,WVEC,PROD)
      IMPLICIT REAL(8) (A-H,O-Z)
      DIMENSION XOPT(*),XPT(NPT,*),BMAT(NDIM,*),ZMAT(NPT,*),D(*), &
        W(*),VLAG(*),S(*),WVEC(NDIM,*),PROD(NDIM,*)
      DIMENSION DEN(9),DENEX(9),PAR(9)

!     N is the number of variables.
!     NPT is the number of interpolation equations.
!     XOPT is the best interpolation point so far.
!     XPT contains the coordinates of the current interpolation points.
!     BMAT provides the last N columns of H.
!     ZMAT and IDZ give a factorization of the first NPT by NPT submatrix of H.
!     NDIM is the first dimension of BMAT and has the value NPT+N.
!     KOPT is the index of the optimal interpolation point.
!     KNEW is the index of the interpolation point that is going to be moved.
!     D will be set to the step from XOPT to the new point, and on entry it
!       should be the D that was calculated by the last call of BIGLAG. The
!       length of the initial D provides a trust region bound on the final D.
!     W will be set to Wcheck for the final choice of D.
!     VLAG will be set to Theta*Wcheck+e_b for the final choice of D.
!     BETA will be set to the value that will occur in the updating formula
!       when the KNEW-th interpolation point is moved to its new position.
!     S, WVEC, PROD and the private arrays DEN, DENEX and PAR will be used
!       for working space.

!     D is calculated in a way that should provide a denominator with a large
!     modulus in the updating formula when the KNEW-th interpolation point is
!     shifted to the new position XOPT+D.

!     Set some constants.

      HALF=0.5D0
      ONE=1.0D0
      QUART=0.25D0
      TWO=2.0D0
      ZERO=0.0D0
      TWOPI=8.0D0*DATAN(ONE)
      NPTM=NPT-N-1

!     Store the first NPT elements of the KNEW-th column of H in W(N+1)
!     to W(N+NPT).

      DO 10 K=1,NPT
   10 W(N+K)=ZERO
      DO 20 J=1,NPTM
      TEMP=ZMAT(KNEW,J)
      IF (J .LT. IDZ) TEMP=-TEMP
      DO 20 K=1,NPT
   20 W(N+K)=W(N+K)+TEMP*ZMAT(K,J)
      ALPHA=W(N+KNEW)

!     The initial search direction D is taken from the last call of BIGLAG,
!     and the initial S is set below, usually to the direction from X_OPT
!     to X_KNEW, but a different direction to an interpolation point may
!     be chosen, in order to prevent S from being nearly parallel to D.

      DD=ZERO
      DS=ZERO
      SS=ZERO
      XOPTSQ=ZERO
      DO 30 I=1,N
      DD=DD+D(I)**2
      S(I)=XPT(KNEW,I)-XOPT(I)
      DS=DS+D(I)*S(I)
      SS=SS+S(I)**2
   30 XOPTSQ=XOPTSQ+XOPT(I)**2
      IF (DS*DS .GT. 0.99D0*DD*SS) THEN
          KSAV=KNEW
          DTEST=DS*DS/SS
          DO 50 K=1,NPT
          IF (K .NE. KOPT) THEN
              DSTEMP=ZERO
              SSTEMP=ZERO
              DO 40 I=1,N
              DIFF=XPT(K,I)-XOPT(I)
              DSTEMP=DSTEMP+D(I)*DIFF
   40         SSTEMP=SSTEMP+DIFF*DIFF
              IF (DSTEMP*DSTEMP/SSTEMP .LT. DTEST) THEN
                  KSAV=K
                  DTEST=DSTEMP*DSTEMP/SSTEMP
                  DS=DSTEMP
                  SS=SSTEMP
              END IF
          END IF
   50     CONTINUE
          DO 60 I=1,N
   60     S(I)=XPT(KSAV,I)-XOPT(I)
      END IF
      SSDEN=DD*SS-DS*DS
      ITERC=0
      DENSAV=ZERO

!     Begin the iteration by overwriting S with a vector that has the
!     required length and direction.

   70 ITERC=ITERC+1
      TEMP=ONE/DSQRT(SSDEN)
      XOPTD=ZERO
      XOPTS=ZERO
      DO 80 I=1,N
      S(I)=TEMP*(DD*S(I)-DS*D(I))
      XOPTD=XOPTD+XOPT(I)*D(I)
   80 XOPTS=XOPTS+XOPT(I)*S(I)

!     Set the coefficients of the first two terms of BETA.

      TEMPA=HALF*XOPTD*XOPTD
      TEMPB=HALF*XOPTS*XOPTS
      DEN(1)=DD*(XOPTSQ+HALF*DD)+TEMPA+TEMPB
      DEN(2)=TWO*XOPTD*DD
      DEN(3)=TWO*XOPTS*DD
      DEN(4)=TEMPA-TEMPB
      DEN(5)=XOPTD*XOPTS
      DO 90 I=6,9
   90 DEN(I)=ZERO

!     Put the coefficients of Wcheck in WVEC.

      DO 110 K=1,NPT
      TEMPA=ZERO
      TEMPB=ZERO
      TEMPC=ZERO
      DO 100 I=1,N
      TEMPA=TEMPA+XPT(K,I)*D(I)
      TEMPB=TEMPB+XPT(K,I)*S(I)
  100 TEMPC=TEMPC+XPT(K,I)*XOPT(I)
      WVEC(K,1)=QUART*(TEMPA*TEMPA+TEMPB*TEMPB)
      WVEC(K,2)=TEMPA*TEMPC
      WVEC(K,3)=TEMPB*TEMPC
      WVEC(K,4)=QUART*(TEMPA*TEMPA-TEMPB*TEMPB)
  110 WVEC(K,5)=HALF*TEMPA*TEMPB
      DO 120 I=1,N
      IP=I+NPT
      WVEC(IP,1)=ZERO
      WVEC(IP,2)=D(I)
      WVEC(IP,3)=S(I)
      WVEC(IP,4)=ZERO
  120 WVEC(IP,5)=ZERO

!     Put the coefficents of THETA*Wcheck in PROD.

      DO 190 JC=1,5
      NW=NPT
      IF (JC .EQ. 2 .OR. JC .EQ. 3) NW=NDIM
      DO 130 K=1,NPT
  130 PROD(K,JC)=ZERO
      DO 150 J=1,NPTM
      SUM=ZERO
      DO 140 K=1,NPT
  140 SUM=SUM+ZMAT(K,J)*WVEC(K,JC)
      IF (J .LT. IDZ) SUM=-SUM
      DO 150 K=1,NPT
  150 PROD(K,JC)=PROD(K,JC)+SUM*ZMAT(K,J)
      IF (NW .EQ. NDIM) THEN
          DO 170 K=1,NPT
          SUM=ZERO
          DO 160 J=1,N
  160     SUM=SUM+BMAT(K,J)*WVEC(NPT+J,JC)
  170     PROD(K,JC)=PROD(K,JC)+SUM
      END IF
      DO 190 J=1,N
      SUM=ZERO
      DO 180 I=1,NW
  180 SUM=SUM+BMAT(I,J)*WVEC(I,JC)
  190 PROD(NPT+J,JC)=SUM

!     Include in DEN the part of BETA that depends on THETA.

      DO 210 K=1,NDIM
      SUM=ZERO
      DO 200 I=1,5
      PAR(I)=HALF*PROD(K,I)*WVEC(K,I)
  200 SUM=SUM+PAR(I)
      DEN(1)=DEN(1)-PAR(1)-SUM
      TEMPA=PROD(K,1)*WVEC(K,2)+PROD(K,2)*WVEC(K,1)
      TEMPB=PROD(K,2)*WVEC(K,4)+PROD(K,4)*WVEC(K,2)
      TEMPC=PROD(K,3)*WVEC(K,5)+PROD(K,5)*WVEC(K,3)
      DEN(2)=DEN(2)-TEMPA-HALF*(TEMPB+TEMPC)
      DEN(6)=DEN(6)-HALF*(TEMPB-TEMPC)
      TEMPA=PROD(K,1)*WVEC(K,3)+PROD(K,3)*WVEC(K,1)
      TEMPB=PROD(K,2)*WVEC(K,5)+PROD(K,5)*WVEC(K,2)
      TEMPC=PROD(K,3)*WVEC(K,4)+PROD(K,4)*WVEC(K,3)
      DEN(3)=DEN(3)-TEMPA-HALF*(TEMPB-TEMPC)
      DEN(7)=DEN(7)-HALF*(TEMPB+TEMPC)
      TEMPA=PROD(K,1)*WVEC(K,4)+PROD(K,4)*WVEC(K,1)
      DEN(4)=DEN(4)-TEMPA-PAR(2)+PAR(3)
      TEMPA=PROD(K,1)*WVEC(K,5)+PROD(K,5)*WVEC(K,1)
      TEMPB=PROD(K,2)*WVEC(K,3)+PROD(K,3)*WVEC(K,2)
      DEN(5)=DEN(5)-TEMPA-HALF*TEMPB
      DEN(8)=DEN(8)-PAR(4)+PAR(5)
      TEMPA=PROD(K,4)*WVEC(K,5)+PROD(K,5)*WVEC(K,4)
  210 DEN(9)=DEN(9)-HALF*TEMPA

!     Extend DEN so that it holds all the coefficients of DENOM.

      SUM=ZERO
      DO 220 I=1,5
      PAR(I)=HALF*PROD(KNEW,I)**2
  220 SUM=SUM+PAR(I)
      DENEX(1)=ALPHA*DEN(1)+PAR(1)+SUM
      TEMPA=TWO*PROD(KNEW,1)*PROD(KNEW,2)
      TEMPB=PROD(KNEW,2)*PROD(KNEW,4)
      TEMPC=PROD(KNEW,3)*PROD(KNEW,5)
      DENEX(2)=ALPHA*DEN(2)+TEMPA+TEMPB+TEMPC
      DENEX(6)=ALPHA*DEN(6)+TEMPB-TEMPC
      TEMPA=TWO*PROD(KNEW,1)*PROD(KNEW,3)
      TEMPB=PROD(KNEW,2)*PROD(KNEW,5)
      TEMPC=PROD(KNEW,3)*PROD(KNEW,4)
      DENEX(3)=ALPHA*DEN(3)+TEMPA+TEMPB-TEMPC
      DENEX(7)=ALPHA*DEN(7)+TEMPB+TEMPC
      TEMPA=TWO*PROD(KNEW,1)*PROD(KNEW,4)
      DENEX(4)=ALPHA*DEN(4)+TEMPA+PAR(2)-PAR(3)
      TEMPA=TWO*PROD(KNEW,1)*PROD(KNEW,5)
      DENEX(5)=ALPHA*DEN(5)+TEMPA+PROD(KNEW,2)*PROD(KNEW,3)
      DENEX(8)=ALPHA*DEN(8)+PAR(4)-PAR(5)
      DENEX(9)=ALPHA*DEN(9)+PROD(KNEW,4)*PROD(KNEW,5)

!     Seek the value of the angle that maximizes the modulus of DENOM.

      SUM=DENEX(1)+DENEX(2)+DENEX(4)+DENEX(6)+DENEX(8)
      DENOLD=SUM
      DENMAX=SUM
      ISAVE=0
      IU=49
      TEMP=TWOPI/DFLOAT(IU+1)
      PAR(1)=ONE
      DO 250 I=1,IU
      ANGLE=DFLOAT(I)*TEMP
      PAR(2)=DCOS(ANGLE)
      PAR(3)=DSIN(ANGLE)
      DO 230 J=4,8,2
      PAR(J)=PAR(2)*PAR(J-2)-PAR(3)*PAR(J-1)
  230 PAR(J+1)=PAR(2)*PAR(J-1)+PAR(3)*PAR(J-2)
      SUMOLD=SUM
      SUM=ZERO
      DO 240 J=1,9
  240 SUM=SUM+DENEX(J)*PAR(J)
      IF (DABS(SUM) .GT. DABS(DENMAX)) THEN
          DENMAX=SUM
          ISAVE=I
          TEMPA=SUMOLD
      ELSE IF (I .EQ. ISAVE+1) THEN
          TEMPB=SUM
      END IF
  250 CONTINUE
      IF (ISAVE .EQ. 0) TEMPA=SUM
      IF (ISAVE .EQ. IU) TEMPB=DENOLD
      STEP=ZERO
      IF (TEMPA .NE. TEMPB) THEN
          TEMPA=TEMPA-DENMAX
          TEMPB=TEMPB-DENMAX
          STEP=HALF*(TEMPA-TEMPB)/(TEMPA+TEMPB)
      END IF
      ANGLE=TEMP*(DFLOAT(ISAVE)+STEP)

!     Calculate the new parameters of the denominator, the new VLAG vector
!     and the new D. Then test for convergence.

      PAR(2)=DCOS(ANGLE)
      PAR(3)=DSIN(ANGLE)
      DO 260 J=4,8,2
      PAR(J)=PAR(2)*PAR(J-2)-PAR(3)*PAR(J-1)
  260 PAR(J+1)=PAR(2)*PAR(J-1)+PAR(3)*PAR(J-2)
      BETA=ZERO
      DENMAX=ZERO
      DO 270 J=1,9
      BETA=BETA+DEN(J)*PAR(J)
  270 DENMAX=DENMAX+DENEX(J)*PAR(J)
      DO 280 K=1,NDIM
      VLAG(K)=ZERO
      DO 280 J=1,5
  280 VLAG(K)=VLAG(K)+PROD(K,J)*PAR(J)
      TAU=VLAG(KNEW)
      DD=ZERO
      TEMPA=ZERO
      TEMPB=ZERO
      DO 290 I=1,N
      D(I)=PAR(2)*D(I)+PAR(3)*S(I)
      W(I)=XOPT(I)+D(I)
      DD=DD+D(I)**2
      TEMPA=TEMPA+D(I)*W(I)
  290 TEMPB=TEMPB+W(I)*W(I)
      IF (ITERC .GE. N) GOTO 340
      IF (ITERC .GT. 1) DENSAV=DMAX1(DENSAV,DENOLD)
      IF (DABS(DENMAX) .LE. 1.1D0*DABS(DENSAV)) GOTO 340
      DENSAV=DENMAX

!     Set S to half the gradient of the denominator with respect to D.
!     Then branch for the next iteration.

      DO 300 I=1,N
      TEMP=TEMPA*XOPT(I)+TEMPB*D(I)-VLAG(NPT+I)
  300 S(I)=TAU*BMAT(KNEW,I)+ALPHA*TEMP
      DO 320 K=1,NPT
      SUM=ZERO
      DO 310 J=1,N
  310 SUM=SUM+XPT(K,J)*W(J)
      TEMP=(TAU*W(N+K)-ALPHA*VLAG(K))*SUM
      DO 320 I=1,N
  320 S(I)=S(I)+TEMP*XPT(K,I)
      SS=ZERO
      DS=ZERO
      DO 330 I=1,N
      SS=SS+S(I)**2
  330 DS=DS+D(I)*S(I)
      SSDEN=DD*SS-DS*DS
      IF (SSDEN .GE. 1.0D-8*DD*SS) GOTO 70

!     Set the vector W before the RETURN from the subroutine.

  340 DO 350 K=1,NDIM
      W(K)=ZERO
      DO 350 J=1,5
  350 W(K)=W(K)+WVEC(K,J)*PAR(J)
      VLAG(KOPT)=VLAG(KOPT)+ONE
      RETURN
      END SUBROUTINE BIGDEN
      
      
      SUBROUTINE BIGLAG (N,NPT,XOPT,XPT,BMAT,ZMAT,IDZ,NDIM,KNEW, &
        DELTA,D,ALPHA,HCOL,GC,GD,S,W)
      IMPLICIT REAL(8) (A-H,O-Z)
      DIMENSION XOPT(*),XPT(NPT,*),BMAT(NDIM,*),ZMAT(NPT,*),D(*), &
        HCOL(*),GC(*),GD(*),S(*),W(*)

!     N is the number of variables.
!     NPT is the number of interpolation equations.
!     XOPT is the best interpolation point so far.
!     XPT contains the coordinates of the current interpolation points.
!     BMAT provides the last N columns of H.
!     ZMAT and IDZ give a factorization of the first NPT by NPT submatrix of H.
!     NDIM is the first dimension of BMAT and has the value NPT+N.
!     KNEW is the index of the interpolation point that is going to be moved.
!     DELTA is the current trust region bound.
!     D will be set to the step from XOPT to the new point.
!     ALPHA will be set to the KNEW-th diagonal element of the H matrix.
!     HCOL, GC, GD, S and W will be used for working space.

!     The step D is calculated in a way that attempts to maximize the modulus
!     of LFUNC(XOPT+D), subject to the bound ||D|| .LE. DELTA, where LFUNC is
!     the KNEW-th Lagrange function.

!     Set some constants.

      HALF=0.5D0
      ONE=1.0D0
      ZERO=0.0D0
      TWOPI=8.0D0*DATAN(ONE)
      DELSQ=DELTA*DELTA
      NPTM=NPT-N-1

!     Set the first NPT components of HCOL to the leading elements of the
!     KNEW-th column of H.

      ITERC=0
      DO 10 K=1,NPT
   10 HCOL(K)=ZERO
      DO 20 J=1,NPTM
      TEMP=ZMAT(KNEW,J)
      IF (J .LT. IDZ) TEMP=-TEMP
      DO 20 K=1,NPT
   20 HCOL(K)=HCOL(K)+TEMP*ZMAT(K,J)
      ALPHA=HCOL(KNEW)

!     Set the unscaled initial direction D. Form the gradient of LFUNC at
!     XOPT, and multiply D by the second derivative matrix of LFUNC.

      DD=ZERO
      DO 30 I=1,N
      D(I)=XPT(KNEW,I)-XOPT(I)
      GC(I)=BMAT(KNEW,I)
      GD(I)=ZERO
   30 DD=DD+D(I)**2
      DO 50 K=1,NPT
      TEMP=ZERO
      SUM=ZERO
      DO 40 J=1,N
      TEMP=TEMP+XPT(K,J)*XOPT(J)
   40 SUM=SUM+XPT(K,J)*D(J)
      TEMP=HCOL(K)*TEMP
      SUM=HCOL(K)*SUM
      DO 50 I=1,N
      GC(I)=GC(I)+TEMP*XPT(K,I)
   50 GD(I)=GD(I)+SUM*XPT(K,I)

!     Scale D and GD, with a sign change if required. Set S to another
!     vector in the initial two dimensional subspace.

      GG=ZERO
      SP=ZERO
      DHD=ZERO
      DO 60 I=1,N
      GG=GG+GC(I)**2
      SP=SP+D(I)*GC(I)
   60 DHD=DHD+D(I)*GD(I)
      SCALE=DELTA/DSQRT(DD)
      IF (SP*DHD .LT. ZERO) SCALE=-SCALE
      TEMP=ZERO
      IF (SP*SP .GT. 0.99D0*DD*GG) TEMP=ONE
      TAU=SCALE*(DABS(SP)+HALF*SCALE*DABS(DHD))
      IF (GG*DELSQ .LT. 0.01D0*TAU*TAU) TEMP=ONE
      DO 70 I=1,N
      D(I)=SCALE*D(I)
      GD(I)=SCALE*GD(I)
   70 S(I)=GC(I)+TEMP*GD(I)

!     Begin the iteration by overwriting S with a vector that has the
!     required length and direction, except that termination occurs if
!     the given D and S are nearly parallel.

   80 ITERC=ITERC+1
      DD=ZERO
      SP=ZERO
      SS=ZERO
      DO 90 I=1,N
      DD=DD+D(I)**2
      SP=SP+D(I)*S(I)
   90 SS=SS+S(I)**2
      TEMP=DD*SS-SP*SP
      IF (TEMP .LE. 1.0D-8*DD*SS) GOTO 160
      DENOM=DSQRT(TEMP)
      DO 100 I=1,N
      S(I)=(DD*S(I)-SP*D(I))/DENOM
  100 W(I)=ZERO

!     Calculate the coefficients of the objective function on the circle,
!     beginning with the multiplication of S by the second derivative matrix.

      DO 120 K=1,NPT
      SUM=ZERO
      DO 110 J=1,N
  110 SUM=SUM+XPT(K,J)*S(J)
      SUM=HCOL(K)*SUM
      DO 120 I=1,N
  120 W(I)=W(I)+SUM*XPT(K,I)
      CF1=ZERO
      CF2=ZERO
      CF3=ZERO
      CF4=ZERO
      CF5=ZERO
      DO 130 I=1,N
      CF1=CF1+S(I)*W(I)
      CF2=CF2+D(I)*GC(I)
      CF3=CF3+S(I)*GC(I)
      CF4=CF4+D(I)*GD(I)
  130 CF5=CF5+S(I)*GD(I)
      CF1=HALF*CF1
      CF4=HALF*CF4-CF1

!     Seek the value of the angle that maximizes the modulus of TAU.

      TAUBEG=CF1+CF2+CF4
      TAUMAX=TAUBEG
      TAUOLD=TAUBEG
      ISAVE=0
      IU=49
      TEMP=TWOPI/DFLOAT(IU+1)
      DO 140 I=1,IU
      ANGLE=DFLOAT(I)*TEMP
      CTH=DCOS(ANGLE)
      STH=DSIN(ANGLE)
      TAU=CF1+(CF2+CF4*CTH)*CTH+(CF3+CF5*CTH)*STH
      IF (DABS(TAU) .GT. DABS(TAUMAX)) THEN
          TAUMAX=TAU
          ISAVE=I
          TEMPA=TAUOLD
      ELSE IF (I .EQ. ISAVE+1) THEN
          TEMPB=TAU
      END IF
  140 TAUOLD=TAU
      IF (ISAVE .EQ. 0) TEMPA=TAU
      IF (ISAVE .EQ. IU) TEMPB=TAUBEG
      STEP=ZERO
      IF (TEMPA .NE. TEMPB) THEN
          TEMPA=TEMPA-TAUMAX
          TEMPB=TEMPB-TAUMAX
          STEP=HALF*(TEMPA-TEMPB)/(TEMPA+TEMPB)
      END IF
      ANGLE=TEMP*(DFLOAT(ISAVE)+STEP)

!     Calculate the new D and GD. Then test for convergence.

      CTH=DCOS(ANGLE)
      STH=DSIN(ANGLE)
      TAU=CF1+(CF2+CF4*CTH)*CTH+(CF3+CF5*CTH)*STH
      DO 150 I=1,N
      D(I)=CTH*D(I)+STH*S(I)
      GD(I)=CTH*GD(I)+STH*W(I)
  150 S(I)=GC(I)+GD(I)
      IF (DABS(TAU) .LE. 1.1D0*DABS(TAUBEG)) GOTO 160
      IF (ITERC .LT. N) GOTO 80
  160 RETURN
      END SUBROUTINE BIGLAG
      
      
      
      SUBROUTINE TRSAPP (N,NPT,XOPT,XPT,GQ,HQ,PQ,DELTA,STEP, &
        D,G,HD,HS,CRVMIN)
      IMPLICIT REAL(8) (A-H,O-Z)
      DIMENSION XOPT(*),XPT(NPT,*),GQ(*),HQ(*),PQ(*),STEP(*), &
        D(*),G(*),HD(*),HS(*)

!     N is the number of variables of a quadratic objective function, Q say.
!     The arguments NPT, XOPT, XPT, GQ, HQ and PQ have their usual meanings,
!       in order to define the current quadratic model Q.
!     DELTA is the trust region radius, and has to be positive.
!     STEP will be set to the calculated trial step.
!     The arrays D, G, HD and HS will be used for working space.
!     CRVMIN will be set to the least curvature of H along the conjugate
!       directions that occur, except that it is set to zero if STEP goes
!       all the way to the trust region boundary.

!     The calculation of STEP begins with the truncated conjugate gradient
!     method. If the boundary of the trust region is reached, then further
!     changes to STEP may be made, each one being in the 2D space spanned
!     by the current STEP and the corresponding gradient of Q. Thus STEP
!     should provide a substantial reduction to Q within the trust region.

!     Initialization, which includes setting HD to H times XOPT.

      HALF=0.5D0
      ZERO=0.0D0
      TWOPI=8.0D0*DATAN(1.0D0)
      DELSQ=DELTA*DELTA
      ITERC=0
      ITERMAX=N
      ITERSW=ITERMAX
      DO 10 I=1,N
   10 D(I)=XOPT(I)
      GOTO 170

!     Prepare for the first line search.

   20 QRED=ZERO
      DD=ZERO
      DO 30 I=1,N
      STEP(I)=ZERO
      HS(I)=ZERO
      G(I)=GQ(I)+HD(I)
      D(I)=-G(I)
   30 DD=DD+D(I)**2
      CRVMIN=ZERO
      IF (DD .EQ. ZERO) GOTO 160
      DS=ZERO
      SS=ZERO
      GG=DD
      GGBEG=GG

!     Calculate the step to the trust region boundary and the product HD.

   40 ITERC=ITERC+1
      TEMP=DELSQ-SS
      BSTEP=TEMP/(DS+DSQRT(DS*DS+DD*TEMP))
      GOTO 170
   50 DHD=ZERO
      DO 60 J=1,N
   60 DHD=DHD+D(J)*HD(J)

!     Update CRVMIN and set the step-length ALPHA.

      ALPHA=BSTEP
      IF (DHD .GT. ZERO) THEN
          TEMP=DHD/DD
          IF (ITERC .EQ. 1) CRVMIN=TEMP
          CRVMIN=DMIN1(CRVMIN,TEMP)
          ALPHA=DMIN1(ALPHA,GG/DHD)
      END IF
      QADD=ALPHA*(GG-HALF*ALPHA*DHD)
      QRED=QRED+QADD

!     Update STEP and HS.

      GGSAV=GG
      GG=ZERO
      DO 70 I=1,N
      STEP(I)=STEP(I)+ALPHA*D(I)
      HS(I)=HS(I)+ALPHA*HD(I)
   70 GG=GG+(G(I)+HS(I))**2

!     Begin another conjugate direction iteration if required.

      IF (ALPHA .LT. BSTEP) THEN
          IF (QADD .LE. 0.01D0*QRED) GOTO 160
          IF (GG .LE. 1.0D-4*GGBEG) GOTO 160
          IF (ITERC .EQ. ITERMAX) GOTO 160
          TEMP=GG/GGSAV
          DD=ZERO
          DS=ZERO
          SS=ZERO
          DO 80 I=1,N
          D(I)=TEMP*D(I)-G(I)-HS(I)
          DD=DD+D(I)**2
          DS=DS+D(I)*STEP(I)
   80     SS=SS+STEP(I)**2
          IF (DS .LE. ZERO) GOTO 160
          IF (SS .LT. DELSQ) GOTO 40
      END IF
      CRVMIN=ZERO
      ITERSW=ITERC

!     Test whether an alternative iteration is required.

   90 IF (GG .LE. 1.0D-4*GGBEG) GOTO 160
      SG=ZERO
      SHS=ZERO
      DO 100 I=1,N
      SG=SG+STEP(I)*G(I)
  100 SHS=SHS+STEP(I)*HS(I)
      SGK=SG+SHS
      ANGTEST=SGK/DSQRT(GG*DELSQ)
      IF (ANGTEST .LE. -0.99D0) GOTO 160

!     Begin the alternative iteration by calculating D and HD and some
!     scalar products.

      ITERC=ITERC+1
      TEMP=DSQRT(DELSQ*GG-SGK*SGK)
      TEMPA=DELSQ/TEMP
      TEMPB=SGK/TEMP
      DO 110 I=1,N
  110 D(I)=TEMPA*(G(I)+HS(I))-TEMPB*STEP(I)
      GOTO 170
  120 DG=ZERO
      DHD=ZERO
      DHS=ZERO
      DO 130 I=1,N
      DG=DG+D(I)*G(I)
      DHD=DHD+HD(I)*D(I)
  130 DHS=DHS+HD(I)*STEP(I)

!     Seek the value of the angle that minimizes Q.

      CF=HALF*(SHS-DHD)
      QBEG=SG+CF
      QSAV=QBEG
      QMIN=QBEG
      ISAVE=0
      IU=49
      TEMP=TWOPI/DFLOAT(IU+1)
      DO 140 I=1,IU
      ANGLE=DFLOAT(I)*TEMP
      CTH=DCOS(ANGLE)
      STH=DSIN(ANGLE)
      QNEW=(SG+CF*CTH)*CTH+(DG+DHS*CTH)*STH
      IF (QNEW .LT. QMIN) THEN
          QMIN=QNEW
          ISAVE=I
          TEMPA=QSAV
      ELSE IF (I .EQ. ISAVE+1) THEN
          TEMPB=QNEW
      END IF
  140 QSAV=QNEW
      IF (ISAVE .EQ. ZERO) TEMPA=QNEW
      IF (ISAVE .EQ. IU) TEMPB=QBEG
      ANGLE=ZERO
      IF (TEMPA .NE. TEMPB) THEN
          TEMPA=TEMPA-QMIN
          TEMPB=TEMPB-QMIN
          ANGLE=HALF*(TEMPA-TEMPB)/(TEMPA+TEMPB)
      END IF
      ANGLE=TEMP*(DFLOAT(ISAVE)+ANGLE)

!     Calculate the new STEP and HS. Then test for convergence.

      CTH=DCOS(ANGLE)
      STH=DSIN(ANGLE)
      REDUC=QBEG-(SG+CF*CTH)*CTH-(DG+DHS*CTH)*STH
      GG=ZERO
      DO 150 I=1,N
      STEP(I)=CTH*STEP(I)+STH*D(I)
      HS(I)=CTH*HS(I)+STH*HD(I)
  150 GG=GG+(G(I)+HS(I))**2
      QRED=QRED+REDUC
      RATIO=REDUC/QRED
      IF (ITERC .LT. ITERMAX .AND. RATIO .GT. 0.01D0) GOTO 90
  160 RETURN

!     The following instructions act as a subroutine for setting the vector
!     HD to the vector D multiplied by the second derivative matrix of Q.
!     They are called from three different places, which are distinguished
!     by the value of ITERC.

  170 DO 180 I=1,N
  180 HD(I)=ZERO
      DO 200 K=1,NPT
      TEMP=ZERO
      DO 190 J=1,N
  190 TEMP=TEMP+XPT(K,J)*D(J)
      TEMP=TEMP*PQ(K)
      DO 200 I=1,N
  200 HD(I)=HD(I)+TEMP*XPT(K,I)
      IH=0
      DO 210 J=1,N
      DO 210 I=1,J
      IH=IH+1
      IF (I .LT. J) HD(J)=HD(J)+HQ(IH)*D(I)
  210 HD(I)=HD(I)+HQ(IH)*D(J)
      IF (ITERC .EQ. 0) GOTO 20
      IF (ITERC .LE. ITERSW) GOTO 50
      GOTO 120
      END SUBROUTINE TRSAPP
      
      
      
      SUBROUTINE UPDATE (N,NPT,BMAT,ZMAT,IDZ,NDIM,VLAG,BETA,KNEW,W)
      IMPLICIT REAL(8) (A-H,O-Z)
      DIMENSION BMAT(NDIM,*),ZMAT(NPT,*),VLAG(*),W(*)

!     The arrays BMAT and ZMAT with IDZ are updated, in order to shift the
!     interpolation point that has index KNEW. On entry, VLAG contains the
!     components of the vector Theta*Wcheck+e_b of the updating formula
!     (6.11), and BETA holds the value of the parameter that has this name.
!     The vector W is used for working space.

!     Set some constants.

      ONE=1.0D0
      ZERO=0.0D0
      NPTM=NPT-N-1

!     Apply the rotations that put zeros in the KNEW-th row of ZMAT.

      JL=1
      DO 20 J=2,NPTM
      IF (J .EQ. IDZ) THEN
          JL=IDZ
      ELSE IF (ZMAT(KNEW,J) .NE. ZERO) THEN
          TEMP=DSQRT(ZMAT(KNEW,JL)**2+ZMAT(KNEW,J)**2)
          TEMPA=ZMAT(KNEW,JL)/TEMP
          TEMPB=ZMAT(KNEW,J)/TEMP
          DO 10 I=1,NPT
          TEMP=TEMPA*ZMAT(I,JL)+TEMPB*ZMAT(I,J)
          ZMAT(I,J)=TEMPA*ZMAT(I,J)-TEMPB*ZMAT(I,JL)
   10     ZMAT(I,JL)=TEMP
          ZMAT(KNEW,J)=ZERO
      END IF
   20 CONTINUE

!     Put the first NPT components of the KNEW-th column of HLAG into W,
!     and calculate the parameters of the updating formula.

      TEMPA=ZMAT(KNEW,1)
      IF (IDZ .GE. 2) TEMPA=-TEMPA
      IF (JL .GT. 1) TEMPB=ZMAT(KNEW,JL)
      DO 30 I=1,NPT
      W(I)=TEMPA*ZMAT(I,1)
      IF (JL .GT. 1) W(I)=W(I)+TEMPB*ZMAT(I,JL)
   30 CONTINUE
      ALPHA=W(KNEW)
      TAU=VLAG(KNEW)
      TAUSQ=TAU*TAU
      DENOM=ALPHA*BETA+TAUSQ
      VLAG(KNEW)=VLAG(KNEW)-ONE

!     Complete the updating of ZMAT when there is only one nonzero element
!     in the KNEW-th row of the new matrix ZMAT, but, if IFLAG is set to one,
!     then the first column of ZMAT will be exchanged with another one later.

      IFLAG=0
      IF (JL .EQ. 1) THEN
          TEMP=DSQRT(DABS(DENOM))
          TEMPB=TEMPA/TEMP
          TEMPA=TAU/TEMP
          DO 40 I=1,NPT
   40     ZMAT(I,1)=TEMPA*ZMAT(I,1)-TEMPB*VLAG(I)
          IF (IDZ .EQ. 1 .AND. TEMP .LT. ZERO) IDZ=2
          IF (IDZ .GE. 2 .AND. TEMP .GE. ZERO) IFLAG=1
      ELSE

!     Complete the updating of ZMAT in the alternative case.

          JA=1
          IF (BETA .GE. ZERO) JA=JL
          JB=JL+1-JA
          TEMP=ZMAT(KNEW,JB)/DENOM
          TEMPA=TEMP*BETA
          TEMPB=TEMP*TAU
          TEMP=ZMAT(KNEW,JA)
          SCALA=ONE/DSQRT(DABS(BETA)*TEMP*TEMP+TAUSQ)
          SCALB=SCALA*DSQRT(DABS(DENOM))
          DO 50 I=1,NPT
          ZMAT(I,JA)=SCALA*(TAU*ZMAT(I,JA)-TEMP*VLAG(I))
   50     ZMAT(I,JB)=SCALB*(ZMAT(I,JB)-TEMPA*W(I)-TEMPB*VLAG(I))
          IF (DENOM .LE. ZERO) THEN
              IF (BETA .LT. ZERO) IDZ=IDZ+1
              IF (BETA .GE. ZERO) IFLAG=1
          END IF
      END IF

!     IDZ is reduced in the following case, and usually the first column
!     of ZMAT is exchanged with a later one.

      IF (IFLAG .EQ. 1) THEN
          IDZ=IDZ-1
          DO 60 I=1,NPT
          TEMP=ZMAT(I,1)
          ZMAT(I,1)=ZMAT(I,IDZ)
   60     ZMAT(I,IDZ)=TEMP
      END IF

!     Finally, update the matrix BMAT.

      DO 70 J=1,N
      JP=NPT+J
      W(JP)=BMAT(KNEW,J)
      TEMPA=(ALPHA*VLAG(JP)-TAU*W(JP))/DENOM
      TEMPB=(-BETA*W(JP)-TAU*VLAG(JP))/DENOM
      DO 70 I=1,JP
      BMAT(I,J)=BMAT(I,J)+TEMPA*VLAG(I)+TEMPB*W(I)
      IF (I .GT. NPT) BMAT(JP,I-NPT)=BMAT(I,J)
   70 CONTINUE
      RETURN
      END SUBROUTINE UPDATE



END MODULE newuoa_module




